Questions tagged [forgetful-functors]

For questions involving the forgetful functor, the functor that drops some of the properties of the input structure before mapping to the output.

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Forgetful functor from Magma to Set

I'm trying to do an exercise on adjoints, where one of the questions asks to prove that the forgetful functor $U: \mathsf{Magma} \rightarrow \mathsf{Set}$ has a left adjoint. Is there a way of ...
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1answer
43 views

Definition of representable functor with values in $Ab$

I was reading these notes and I came up with a question about Definition 8.2.1 at page 283: what does it mean for a functor $F : C \to Ab$ to be representable? Is this definition related to that of a ...
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102 views

The forgetful functor $U: \text{Con}(\mathcal C) \to \mathcal C$ has a left adjoint.

Let $\mathcal C$ a category. We define the category of cones $\text{Con}(\mathcal C)$ in the following way: Objects: quadruples $(\mathcal Z, \underline{M}, M, m)$ consisting of a small category $\...
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1answer
179 views

Reconstructions of Groups From Category of $G-\mathbf{Sets}$; Construction of a Group Homomorphism [duplicate]

I try to come up with a proof of the following statement, but I find it a little difficult. I hope that I can get some help from someone on this site. I think this is what they give a proof of, on ...
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concrete functor in accessible categories

What is a concrete functor $U$ in the context of accessible categories. Everyvere used, never defined. From where to where it leads ?
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1answer
21 views

construction of adjoint of forgetful functor Set_\star to Set

I need to determine if the forgetful functor \begin{equation} U: Set_\star \longrightarrow Set \end{equation} that forgets "the base points" has left adjoint or right adjoint, but I'm ...
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1answer
58 views

Naturality of $\varphi : \textbf{Vct}_K(V(x), w) \xrightarrow{\sim} \textbf{Set}(x, U(w))$ in the variable $x$ (Cats for the Working Mathematician).

Consider the forgetful functor $U : \textbf{Vct}_K \to \textbf{Set}$ and the functor $V : \textbf{Set} \to \textbf{Vct}_K$ that takes an object $x$ in set to the $K$-vector space $V(x)$ with basis $x$ ...
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1answer
65 views

The forgetful functor $U:\mathbf{B}G\to\mathbf{Sets}$ need not preserve infinite limits.

This is Exercise I.7 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". Here $\mathbf{B}G$ is the category of all continuous $G$-sets, where $G$ is a topological group. The ...
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2answers
75 views

Self-Natural Maps for Forgetful Functor

Let $\mathscr{F}$ denote the forgetful functor from the category of groups to the category of sets. Why is there more then one natural map from $\mathscr{F}$ to $\mathscr{F}$? What are all of the ...
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1answer
75 views

Is direct limit created by the forgetful functor from an Eilenberg-Moore algebra

Proposition. Let $T$ be a monad on $C$ and consider the forgetful functor $$ R^T \colon C^T \to C $$ from the category of Eilenberg-Moore algebras to $C$. This functor creates limits; creates ...
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1answer
87 views

Mac Lane: Forgetful functor of Algebraic Systems to $\textbf{Set}$

In Mac Lane's Catergories for the Working Mathematcian text, he introduces the concepts of universal algebra to explain why the forgetful functors $\textbf{Grp} \to \textbf{Set}$, $\textbf{Ab} \to \...
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1answer
73 views

Forgetful functor from Pos to Set does not have a right adjoint

I'm trying to show that the forgetful functor from $Pos$ to $Set$ does not have a right adjoint, by showing that it does not preserve coequalizers. The hint in the lecture notes I am studying, ...
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1answer
87 views

How does the forgetful functor from $\mathbf{C}/C$ to $\mathbf{C}$ forgets the object $C$?

First, sorry for duplication. I've noticed How does the functor $F: \textbf{C}/C \to \textbf{C}$ "forget about the base object" $C$?, but answers there didn't solve my confusion, and my ...
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Representability of functors in categories other than Set category

Let $\mathcal C,\mathcal D$ be locally small categories and also assume that $\mathcal D$ is small and that every morphism in $\mathcal D$ is a function between sets. Assume that for every $A,B \in \...
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562 views

Why is the forgetful functor representable?

I'm reading Adowey's Category Theory, and I'm struggling with the last exercise of the second chapter, which is to show that the forgetful functor for monoids, $U : \mathbf{Mon} \to \mathbf{Sets}$, is ...
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3answers
104 views

Is a forgetful functor necessarily unfaithful or non-isomorphic?

Can forgetful functor be defined accurately? I feel the wikipedia article and Categories for the Working Mathematician don't define the concept rigorously. Is a forgetful functor necessarily ...
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1answer
95 views

2-functor and CAT

Let $\cal K$ be a small category. Let $\cal A$ be a subcategory of $\mathbf {CAT}$ and $U:{\cal A}\hookrightarrow{\mathbf {CAT}}$ the underlying functor. Now how is $U^{\cal K}$ naturally defined as a ...
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1answer
220 views

Completeness of over category (slice)

Let $\mathcal J$ be a (small) category (denote $I:= \mathcal J_0$) and $\mathcal C$ a category that has all (small) limits (all limits of shape $\mathcal J$ for all $\mathcal J$). Prop 3.4 states then ...
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3answers
125 views

Why does $\mathbb{Z}$ represent the forgetful functor $U:\mathbf{Grp}\to\mathbf{Set}$

This is from Emily Riehl's Category theory in context The forgetful functor $U:\mathbf{Grp}\to\mathbf{Set}$ is represented by the group $\mathbb{Z}$ thanks to the natural isomorphism $\alpha:\mathbf{...
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1answer
75 views

Do subobjects in concrete categories correspond to subsets?

A concrete category is a category $C$ endowed with a faithful functor $U:C\rightarrow Set$. And if $a$ is an object in $C$, then a subobject of $a$ is an isomorphism class of monomorphisms with ...
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137 views

Is there a notion of a transversal of subobjects?

If $a$ is an object in a category $C$, then a subobject of $a$ is an isomorphism class of monomorphisms with codomain $a$. The subobjects of all the objects in $C$ partitions the class of ...
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Is the category of small graphs a concrete category?

Let $\mathbf{Grf}$ be the category whose objects are small graphs and whose arrows are graph homomorphisms. A small graph is tuple $G = \langle V, E, \operatorname{src}, \operatorname{trg}\rangle$ ...
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1answer
143 views

Composition of monadic functors isn't monadic

Disclaimer: this question already has a solution here: Composition of monadic functors may not be monadic . However, I would like to understand how to solve this using another characterisation of ...
2
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1answer
108 views

$\lambda$-accessible categories, unclear proof

In the context of $\lambda$-accessible categories consider the proof of the proposition $1.22$ here. How/where did we use the fact that $\cal D$ in the last but one line has less than $\lambda$ ...
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2answers
157 views

Underlying set of the free monoid, does it contain the empty string?

In the free monoid over a set the unique sequence of zero elements, often called the empty string is the identity element. Is the empty string an element of the underlying set of the free monoid? In ...
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1answer
115 views

How to see that endotransformations of fiber functor have a coalgebra structure?

This question is based on section 5.2 in Tensor Categories, by Etingof et al. Note also that the question is pretty much in the title and what follows is just some background along with my fruitless ...
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45 views

Subfunctor from a category to Set

I'm looking for the definition of a subfunctor $R$ of a functor $U:\cal K \to \mathbb {Set}$. Are there any other conditions beyond that $R(A)\subseteq U(A)$ for all objects $A\in \cal K$ ?
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2answers
156 views

Forgetful functor from complete metric space to metric space

I am reading Mac Lane's Categories for the Working Mathematician. He mentioned that the usual completion of metric space is universal for the evident forgetful functor (from complete metric spaces to ...
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1answer
146 views

A formal definition of forgetful functor

I have read in many category theory textbooks the term "forgetful functor". But no has ever given me a precise definition of this term. I want a rigorous definition, not merely an answer that ...